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Scalable numerical methods for partial differential equations

		An overlapping domain, on which the partial differential equation is discretized using a low-precision method, is built on top of the high-order element of the GASpAR code. The special boundary conditions applied to this overlapping element will reduce the overall time to solution of the solver.

In FY2007, work focused on the GASpAR geophysical and astrophysical spectral element adaptive code. Major code updates needed to be performed to support the special communication necessary in Schwarz and optimized Schwarz methods. This was completed and tested during FY2007. Also, since the optimized Schwarz preconditioning is a non-symmetric method, a new Krylov subspace method was required. The BiCGStab method was chosen because it produced satisfactory results for the HOMME model and the optimized Schwarz technique. Also accomplished in FY2007 is the inclusion of standard Schwarz preconditioning to test the new features of the code. It is based on the work of Fisher in 1997. A low-order Q1 finite element is constructed on an overlapping grid.

Our intent for FY2008 is to complete the optimized Schwarz work for incompressible MHD equations with a special coarse solver based on radial basis functions that seem to offer flexibility in an adapted mesh environment. The work for the POP ocean model will be postponed until the technology is sufficiently mature in the GASpAR code.

This project supports NCAR's strategic priorities of "Conducting research in computer science, applied mathematics, statistics, and numerical methods," "Developing community models," and "Improving prediction of weather, climate, and other atmospheric phenomena." It is made possible through NSF Core funding.